where \(\tilde{\mathbb {E}}()\)  means expectation with respect to \(\tilde{\mathbb {P}}\)Formula (REF ) is commonly known as the Kallianpur-Striebel formula. For a proof, see, e.g., Prop 3.16 in [1]}.. Since, under \(\tilde{\mathbb {P}}\) , \(X\)  and \(Y\)  are independent we can interpret the numerator \(\tilde{\mathbb {E}}\left(f((X_s)_{s\in [0,t]})~Z_t(X,Y)~\vert ~ {\cal F }_t\right)\)  and
\(\tilde{\mathbb {E}}\left(Z_t(X,Y)~\vert ~ {\cal F }_t\right)\)  as integrals with respect to the law of the signal, where the integrant has the observation process path fixed \(Y\) .This interpretation can be made rigorous, see e.g. the robust representation formulation of the filtering problem described in Chapter 5 in [1]}.. We can therefore, let \(\overline{Z}_t(Y)\)  be the likelihood function defined by
\(\overline{Z}_t(Y):=\mathbb {E}_Y\left(Z_t(X,Y)\right),\) 
